Should I model a problem with quantised output as classification or regression?
So this is considered Ordinal Regression. There are many ways to model this type of data, generally in some form of regression setting. I do not recommend the softmax route because as you mentioned, there exists prebuilt correlation to the outputs.  

Some common ways to approach this
(Note that im assuming your looking for methodologies that can be optimized through gradient techniques because of the way you formulated your question)  


Treat it as a normal regressor where you clip the output to your range, and then define thresholds arbitrarily, such as $.36 \rightarrow .4$, $.34 \rightarrow .3$, etc..
Use a bounded activation function then scale (if your outputs are [0,1,.1], you could use sigmoid and just scale by a factor of 1, but if [0,10,1], you could use sigmoid and scale by 10. Once again youll need to create arbitrary threshold between each 2 points for inference (this can also be incorporated into your loss)  
Using the above two methods, but learn the threshold between each 2 ordered points. Have an output layer that learns the ideal threshold for inference  


And if you didnt want to use neural netowrks, each of these approaches have a bayesian analog that work as well!